Nested-sequent Calculus for Modal Logic MB

Abstract: Quantum logic (QL) is a non-classical logic for analyzing the propositions of quantum physics. Modal logic MB, which is a logic that handles the value of the inner product that appears in quantum mechanics, was constructed with the development of QL. Although the basic properties of this logic have already been analyzed in a previous study, some essential parts still need to be completed. They are concerned with the completeness theorem and the decidability of the validity problem of this logic. This study solves those problems by constructing a nested-sequent calculus for MB. In addition, new logic MB+ with the addition of new modal symbols is discussed.